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You go a long way from elementary school in which they ask you what 3+5 equals, and without a doubt and without hesitation, you shout "Eight!"

So what's changed since then? Concepts have been added to mathematics which blur the solidity that we once had as children. Ironically, the more we know of mathematics, the less certain we are.

Almost certainly one of the first instances of this concept growing up were negative numbers. The concept is abstract, and difficult for a child to grasp, as you can never have a negative amount of apples. However, you realize that it is merely a convenience in mathematics to be able to measure in the amount of apples you'd have to have taken away to have no more apples.

And in middle school, we learn that there are some numbers which cannot be calculated with a calculator. Does anyone really know what 1/0 equal? Undefined is not a number value, but a way to uncomplicate a complicated subject. Most high level mathematicians claim that 1/0 is positive infinity, though a handful would suggest it might also be negative infinity as positive infintiy and negative infinity wrap. What's more, it'll never be proven. Infinity is an impossible scenario.

So what else don't we know? Imaginary numbers are not supposed to really exist, but they are used commonly in predicting circuitry behavior by engineers. And, as a matter of fact, would make the task virtually impossible without the existence of imaginary numbers.

About an abstract concept as you can get that we take advantage of everyday is the number zero. It's considered to be a mathematical achievement by civilizations to grasp the concept of zero and usually a breaking point in scientific developments in a civilization. It's one of the easiest grasped concepts, though at one time it wasn't. Why would you use a symbol to represent nothing? Like negative numbers, you cannot have negative apples, so why have a symbol to respresent no applies?

Then you break into territory of Godel's Incompletness theorem which states that you can prove the functionality of number theory to a finite number. However, the set of all integer numbers in theory does not end, and so for any finite number we can prove to work in number theory, there is that number + 1 which remains unproven.

We forget that mathematics is founded on observation, albeit a very persistent one. The basis for any calculation done in mathematics leans upon principles built upon fundamental axioms, which have NOT been proven and furthermore can never be proven. This is different than logic, which remains universal because we define that true is not false and false is not true, then by our own definition, you cannot ever show that true is equal to false. This is true by definition, not by observation. Therefore, a triangle could never have four sides, for instance.

However, mathematics is logic based on axioms which are not proven. Take for instance the expression 0 + x = x. This is based on the axiom which states that if you have an object and you add nothing to it, you will have only the object remaining. This is not a definition but an almost trivial observation. The difference being that an observation could potentially change tomorrow.

Though there is nothing in mathematics that can prove to us that tomorrow it won't simply change. In another part of the universe, these axioms might truly be different. Just because we cannot imagine a world which behaves differently than our own doesn't mean that it doesn't exist. Perhaps in a part of the universe, 0 + x = c * x, where c is a mathematical constant. The beings which live there would be godly, using technology to create anything from nothing. The fact is we simply don't know, which brings me back to my original point. What do we really know of mathematics? If it is not potentially universal, then it is no different really than a theory with lots of supporting empirical evidence to support it.

Then really any future developments in mathematics we can expect to continue this pattern of ambiguity and uncertainty until we begin realize that perhaps because these principles truly are founded on observation, and not proven fact, and that we've been wrong about a lot of things. Only then can we hope to discover the true nature of mathematics.

If the world should blow itself up,the last audible voice would be an expert saying it can't be done

Then really any future developments in mathematics we can expect to continue this pattern of ambiguity and uncertainty until we begin realize that perhaps because these principles truly are founded on observation, and not proven fact, and that we've been wrong about a lot of things. Only then can we hope to discover the true nature of mathematics.

This is true in many different fields of study; things that we absolutely KNOW today will be disproven or called ridiculous 30 years from now. Though some of our means of identifying with our sense of reality, like using mathematics to define indescribable phenomena, may be laughable in the future, what it does is provide a path to where we will be.

Beware the following generalization:

At any point in history, when man called his time 'Modern' he thought of himself as the ruler of his universe with the best of the best knowledge that best described his observable world. We are now at a point where we are beginning to admit openly that we do not know everything, that we are less than infants as beings in this universe, and that we have many eons to go before completely understanding the nature of all things.

That's great news for people that are aggrevated with our current state of knowledge concerning the sciences and its inability to truly reflect the attainable, or the complete nature of all that is, because that means that we are on a path of interest that may eventually lean our species in the right path of knowledge-gathering. That the wisdom that we sorely desire may not be attained during our lifetimes purely sucks for us as individuals, we can feel becalmed in the fact that imaginary numbers and chaotic influences in mathematics may, eventually, answer some of the questions that we are just learning to ask about our universe.

I am just me, searching for simplicity.........and a good hair stylist